A number is divisible by three iff the sum of its digits is divisible by three.
First, prove 10^N = 1 mod 3 for all integers N >= 0.
1 = 1 mod 3. 10 = 1 mod 3. 10^N = 10^(N-1) * 10 = 10^(N-1) mod 3.
QED by induction.
Now let D0? be the units digit of N, D1? the tens digit, etc.
Now N = Summation From k=0 to k=inf of Dk?*10^k.
Therefore N mod 3 = Summation from k=0 to k=inf of Dk? mod 3. QED